Knowing_and_Teaching_Elementary_Mathemat (1)

Subtraction with Regrouping: Approaches to Teaching a Topic
Eighty-six percent of the Chinese teachers described the “taking” step in the algorithm
as a process of “decomposing a unit of higher value.” Instead of saying that “you borrow 1
ten from the tens place,” they said that “you decompose 1 ten.” 1
The reason that one cannot compute 21–9 directly lies in the form of the number 21. In
the decimal system, the numbers are composed according to the rate of 10. Given that a
number gets 10 units at a certain place value (e.g., ones place or tens place), the 10 units
should be organized into 1 unit of the next higher place value (e.g., tens place or hundreds
place). Theoretically, no more than 9 “scattered” (uncomposed) units exist in the decimal
number system. Now we want to subtract 9 scattered ones units from 21. The latter only has
1 ones unit. The solution, then, is to decompose a unit of higher value, a 10, and subtract 9
individual ones units from the recomposed 21.
During the interviews, the teachers tended to discuss the idea of “decomposing a higher
valued unit” in connection to addition with carrying—“composing a unit of higher value
[jin yi].” In describing how she would teach this topic, Tr. L., an experienced teacher who
teaches first through third grades, said:
I would start with a problem of straightforward subtraction, like 43−22=? After they
solve it, I would change the problem into 43−27=? How does the new problem differ
from the first one? What will happen when we compute the second problem? They
will soon find that 7 is bigger than 3 and we do not have enough ones. Then I would
say, OK, today we don’t have enough ones. But sometimes we have too many ones.
You must remember that last week when we did addition with carrying we had too
many ones. What did we do at that time? They will say we composed them into tens.
So when we have too many ones, we compose them into tens, what can we do when
we don’t have enough ones? We may decompose a 10 back to ones. If we decompose
a 10 in 40, what will happen? We will have enough ones. In this way I will introduce
the concept of “decomposing a unit of higher value into 10 units of a lower value.”
Some teachers indicated that the term “decomposing” suggests its relationship with the
concept of “composing.”
How come there are not enough ones in 53 to subtract 6? Fifty-three is obviously
bigger than 6. Where are the ones in 53? Students will say that the other ones in 53
have been composed into tens. Then I will ask them what can we do to get enough
ones to subtract 7. I expect that they will come up with the idea of decomposing a 10.
Otherwise, I will propose it. (Tr. P.)
In China, as in the United States, the term “borrowing” used to be a traditional metaphor
in subtraction. 2 Ms. S., a third-grade teacher in her second year of teaching, explained why
she thought that the concept of “decomposing a higher value unit” made more sense than
the metaphor of borrowing:
7
1
This aspect has also been observed by other scholars. Stigler and Perry (1988a) reported that
C hinese teachers emphasize “the composition and decomposition of numbers into groups of ten.”
2
Early versions of modern Chinese arithmetic textbooks used the term “subtraction with borrowing”
translated from the West. During the past few decades, the textbooks have used instead
“subtraction with decomposing.”